A proof of Smale's mean value conjecture

Dual Smale's mean value conjecture for $n=7$

We give an analytic proof of the dual Smale's mean value conjecture in the case $n=7$.

On Smale's Mean Value Conjecture

In this paper, we prove Smale's mean value conjecture by making use of quasiconformal deformations and holomorphic motions.

Smale's mean value conjecture for finite Blaschke products

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.

奇多项式的对偶 Smale 平均值猜想

Dual Smale's mean value conjecture for odd polynomials

本文证明了所有线性项非零的奇多项式均满足对偶 Smale 平均值猜想，并给出临界点处的明确下界，构成 T. W. Ng 相关定理的对偶结果。

斯梅尔中值猜想及其关于复多项式的对偶猜想

Smale's Mean Value Conjecture and its Dual Conjecture for Complex Polynomials

本文针对复多项式的斯梅尔中值猜想及其对偶猜想展开研究，通过改进Koebe型定理与马尔可夫型不等式，将对偶猜想的最优无条件下界从1/d²提升至(d−1/2)^(1/d)/d²。

Mean Value Conjectures for Rational Maps

Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p'(z)| in terms of the gradient of a chord from (z, p(z)) to some stationary point on the graph of $p$. The conjecture does not immediately generalise to rational maps since its formulation is invariant under the group of affine maps, not the full Mobius group. Here we give two possible generalisations to rational maps, both of which are Mobius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidat…